2.72 elements of mechanical design

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2.72 Elements of Mechanical Design Spring 2009

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2.72 Elements of

Mechanical Design

Lecture 06: Constraints

Schedule and reading assignment Quiz

� Quiz – HTMs

Topics � Principles of exact constraint� Bearing layout exercises

� Spindle shaft constraint/bearing layout

Reading assignment � Chapter 11 in Shigley and Mischke

• Read sections 11.1 – 11.6, 11.9 • Skim sections 11.7 – 11.8, 11.10 – 11.12

� This is 40ish pages, but most of it is pictures/graphs/examples

© Martin Culpepper, All rights reserved 2

Principles of exact constraint

Under, exact and over constraint Constraints are fundamental to mechanical design

� A mechanical designers goal is to control, i.e. to constrain, parts so that they are where they are supposed to be.

Exact constraint: � There should be one constraint for each degree-of-freedom that is

constrained.

Under constraint � Too few constraints, part is not held in all the directions it needs to be

Over constraint � Too many constraints, some constraints may fight each other when

trying to do the same job.


Mechanical constraints We want to learn how to model and design each

We first need to know: � How they should be used

� What their functional requirements are

How they should be used and their FRs depend upon:� How they are laid out� Dos and don’ts

Learn to lay them out right � Use this to obtain their Functional Requirements

� Then do the detailed design of each


Constraints and Degrees-of-freedom Rigid components have 6 degrees-of-freedom

We will represent an ideal constraint as a line

6 – C = R

C = # of Non-Redundant Constraints

R = # of Independent Degrees of Freedom

Example:

6 - 2constraints = 4DoFs

Rz

Courtesy John Hopkins, MIT


Constraints, compliance and motion Exact constraint: Achieve desired motion

� By applying minimum number of constraints y � Arranging constraints in correct constraint topology � Adding constraints only when necessary z

For now: � Start with ideal constraints

x

Stage Ground

� Considering small motions

� Constraints = lines

Focus on rigid stage attached to ground � What do we mean by rigid?

� What do we mean by constraint? - Stiffness ratios


The benefits of exact constraint

Penalties for over constraint © Martin Culpepper, All rights reserved

Parts of machines are now always the same strength and stiffness.

Large, stiff components have a tendency to “kill” their smaller counterparts when they are connected so that they are forced to fight.

Exact constraint design helps to prevent fights, therfore all your parts live in harmony.

8

X

Y

A

B

A

B

Loose

Binding

Figure by MIT OpenCourseWare.

Constraint examples

1 2

More examples: y

3 z x

4 5 6


Exact constraint practice

Mechanical constraints: Some bearing types Sliding

� Bushings

� etc…

Radial rolling � Radial

� Shallow groove

� Deep groove (Conrad)

� Angular contact

� Tapered © Martin Culpepper, All rights reserved 11

Examples drawn from your lathe


Example: Carriage constraint x

z

y

Assume these are bushings as in

your lathe


Example cont.y

x z

Misalignment& assembly

errors


Example cont. x

z

y

Thermally Induced growth


x

Practical embodiment y

z

Flexure that is stiff in y and z, yet compliant in x


Group exercise – Carriage constraints Identify the motions that you desire for the carriage and

the minimum # of constraints that are needed to yield only these motions.

Identify the constraints from each bearing set and determine how they act in concert to yield the desired motions.


Constraint layouts and

thermal stability

Housing

Avoiding over constraint How to deal with the thermal growth issue

� The shaft typically gets hotter than the housing because the housing has better ability to carry heat away

� Whether the outer or inner race are fixed, matters…

Constraining front and rear bearings� One bearing set should be axially and radially restrained � The other bearing set should ONLY have radial restraint

19© Martin Culpepper, All rights reserved

Shaft Chuck

NutGear

Part

Tool

8.0 04.010.0

Gear

Housing

Examples: Good or bad Outer race fixed axially, if shaft heats is this bad?

� Think about what happens to the preload…

Shaft Chuck


Housing

Examples: Good or bad Outer race fixed axially, if shaft heats is this bad?

� Think about what happens to the preload…

Shaft Chuck

This is the back-to-back config.

It should be used when the outer race is not rotating


Housing

Examples: Good or bad Assume the outer

race does not rotate

Shaft Chuck

Nut

Gear Gear

Sliding permitted

Spacer

1


Housing



Shaft Chuck

Nut

Gear Gear

Sliding permitted

Spacer

2


Housing

Examples: Good or bad

Shaft Chuck

Nut

Gear Gear

Assume the outer race does not rotate

3


Housing

Examples: Good or bad

Shaft Chuck

Nut

Gear Gear

Spring


4


Housing



Shaft Chuck

Nut Gear Gear

5


Housing


race does not rotateGood � Front set constrained radially and axially � Rear set constrained only radially

Shaft Chuck

Nut

Gear Gear

Sliding permitted

Spacer

1

BUT, sliding means some gap must exist and therefore one must precision fabricate if a small gap is desired


Housing


race does not rotateBad � Front set constrained only radially � Rear set constrained only radially

Shaft Chuck

Nut

Gear Gear

Sliding permitted

Spacer

2

This design will not work if axial loads are to be applied along both directions


Housing


race does not rotateGood � Front set constrained radially and ½ axially � Rear set constrained radially and ½ axially

Shaft Chuck

Nut

Gear Gear

3

BUT, adding a spring increases part count/cost

Axial stiffness values (left vs. right) will be different © Martin Culpepper, All rights reserved 30

Housing


race does not rotateBAD � Front set is constrained only ½ axially � Rear set is constrained only ½ axially

Shaft Chuck

Nut

Gear Gear

Spring

4

This design will not work if axial loads are to be applied along both directions


Housing


race does not rotateBAD � Front set is constrained ½ axially and radially � Rear set is constrained ½ axially and radially

Shaft Chuck

Nut Gear Gear

5

At high speeds/loads, thermal growth may kill the bearing sets � Like a double face-to-face…


Housing

Examples: Good or bad Bad

� Front set constrained radially and ½ axially

� Rear set constrained radially (if flexure is of proper stiffness)


GearGear

Shaft Chuck

6

Left bearing set is not in the back-to-back configuration

Shaft can pop out… © Martin Culpepper, All rights reserved 33

Group exercise

Group exercise – Spindle constraints The spindles you have seen use tapered roller bearings.

First, sketch a layout from one of the previous lathes and diagnose its layout

Second, generate and sketch a different way to constrain the spindle shaft.


2.72 elements of mechanical design

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